Topology-based Methods for Analysis and Visualization of Noisy Data

基于拓扑的噪声数据分析和可视化方法主要研究者：Bernd Hamann，加州大学戴维斯分校摘要由不断发展的超级计算机、大型传感器网络和高分辨率成像设备产生的科学数据集的大小正以指数速度迅速增加。该项目解决了对更有效的数据分析方法的需求。它开发了与分析和表示非常大的科学数据集有关的技术，强调捕捉定性特征的概念。鉴于直接应用于“原始”科学数据集的纯粹可视化方法的局限性，该项目旨在为非常大和复杂的数据集的可视化设计新的概念。正在开发的方法首先从给定的数据集中提取有意义的定性信息，然后将其用于以更紧凑的形式呈现数据集的更高级别的信息内容，从而强调相关的定性行为。该项目建立在经典拓扑和几何的概念基础上，其对计算拓扑学和计算几何学的相对新的领域的发展做出了实质性的贡献。这两个领域具有巨大的潜力，大大推进了可视化技术，以了解非常大的，复杂的数据集。该项目适应（和推广）计算拓扑和计算几何算法，这些算法是为真实世界的光滑数学函数建立的，有限样本数据集，即，在有限数量的点上采样的函数（可能通过网格连接）。现实世界的数据集是嘈杂的，这进一步复杂化的拓扑方法，最初开发的光滑函数的应用。本项目研究了基于莫尔斯和莫尔斯-斯梅尔理论（研究临界点行为并得出有关函数的定性结论）的技术的推广，以离散化随时间变化并包含噪声的标量场。

Topology-based Methods for Analysis and Visualization of Noisy DataPrincipal Investigator: Bernd Hamann, University of California, DavisAbstractThe size of scientific data sets that are generated by evolving supercomputers, large sensor networks, and high-resolution imaging devices is increasing rapidly, at an exponential rate. This project addresses the need for more effective data analysis methods. It develops technologies concerned with the analysis and representation of very large scientific data sets, emphasizing concepts that capture qualitative characteristics. In light of the limitations of purely visualization-based approaches applied to "raw" scientific data sets directly, this project aims at devising new concepts for visualizing very large and complex data sets. The methods being developed first extract meaningful qualitative information from a given data set, which is then used to present the higher-level information content of the data set in a significantly more compact form, thus stressing relevant qualitative behavior.The project builds on concepts from classical topology and geometry, which have contributed substantially to the development of the relatively new fields of computational topology and computational geometry. These two fields hold great potential for substantially advancing the visualization technology for understanding extremely large, complicated data sets. This projects adapts (and generalizes) computational topology and computational geometry algorithms that are well-established for smooth mathematical functions to real-world, finite-sample data sets, i.e., functions sampled at a finite number of points (that could possibly be connected by a mesh). Real-world data sets are noisy, which further complicates the application of topological methods that were developed originally for smooth functions. This project investigates the generalization of techniques based on Morse and Morse-Smale theory (studying critical-point behavior and drawing qualitative conclusions about functions) to discretized scalar fields that change over time and also contain noise.